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Fundamental Journal of Modern Physics
Vol. 8, Issue 1, 2015, Pages 1-14
Published online at http://www.frdint.com/
:esphras and Keywords mass transport, diffusion equation, diffusion into silicon, Laplace’s
integral transform, diffusion of Arsenic into Silicon.
*Corresponding author
Received December 19, 2014
© 2015 Fundamental Research and Development International
DIFFUSION IN SEMICONDUCTORS BY USING LAPLACE’S
INTEGRAL TRANSFORM TECHNIQUE
M. K. EL-ADAWI1,*
, S. E.-S. ABDEL-GHANY2 and S. A. SHALABY
1
1Physics Department
Faculty of Education
Ain Shams University
Heliopolis, Cairo, Egypt
e-mail: adawish1@hotmail.com
Safaashalaby16@hotmail.com
2Physics Department
Faculty of Science
Benha University
Benha, Egypt
e-mail: saeedkmghany@yahoo.com
said.alsayed@fsc.bu.edu.eg
Abstract
A theoretical approach to study diffusion in semiconductors is introduced.
A mass-energy model for diffusion atoms into target materials has been
built up. The diffusion equation written in terms of the incident atom
current density (fluence) is introduced. Laplace’s Integral Transform
Technique is applied to get the solution. The concentration function is
obtained using Fick’s first law that relates the mass transport with the
concentration gradient, together with a flux balance equation.
Computations for the case of time-independent incident atomic flux of
different values of Phosphorus, Gallium, Indium and Arsenic diffused into
EL-ADAWI, ABDEL-GHANY and SHALABY
2
Silicon target material are given as illustrative examples. Results show that
the penetration depth of Phosphorus atoms into Silicon is much greater
than that for Indium atoms, while the concentration atoms of Phosphorus
atoms in Silicon is much less than that for Indium atoms. The same
behavior is shown with respect to Arsenic and Gallium atoms diffused in
Silicon target.
1. Introduction
It is well known that the selective introduction of impurities (dopants) into
semiconductors and most materials by diffusion or ion implantation leads to changes
in their chemical, physical, and electronic properties [1, 2]. The structure and the
electrical conductivity may also be modified.
This topic has aroused the interest of many researchers due to its important
applications, such as in the production of electronic devices.
Doping by diffusion is still one of the acceptable processes that have important
technological applications [3, 4], such as the fabrication of the integrated circuits (IC)
that forms the basis of many of the solid state device technology.
It is also useful in the preparation of alloys which possess low surface energy in
the metallic surface [5]. This makes it possible to realize dropwise condensation
mode on such surfaces, which has vital technological applications. Doping permits
the freedom to design the desired activation energy of a dopant in a semiconductor
[6]. Silicon is the main n-type dopant used in GaAs, and it is usually incorporated
into GaAs by ion implantation or by diffusion employing an external source [7].
In practice, the diffusion process consists of deposition followed by diffusion
(driven-in process). The development of the theoretical studies of atomic diffusion is
still of great interest [8-11].
The diffusion of Boron and Phosphorus into Silicon has been investigated by the
statistical moment method (SMM). A mass transfer model for doping a metal target
at elevated temperatures has been built up [12] based on transport of ions in matter
and radiation enhanced diffusion. The process was simulated by a dynamic Monte
Carlo (MC) method to calculate the concentration-depth profiles.
In the present trial, single step diffusion is considered, where a semiconductor
substrate is exposed to high concentrations of the desired impurity exceeding the
level required to achieve solid solubility of the dopant at the semiconductor surface.
DIFFUSION IN SEMICONDUCTORS BY USING LAPLACE’S …
3
The aim of the present trial is to solve the parabolic diffusion equation written in
terms of the current density ( )txJ , of atoms rather than the concentration, together
with Fick’s first law and a flux balance equations. The solution is obtained
analytically using Laplace integral transform technique. As illustrative examples, the
obtained concentration-depth functions are computed for the diffusion of Phosphorus,
Gallium, Indium and Arsenic atoms into Silicon target material. Comparison between
these cases is also considered.
2. Mathematical Formulation of the Problem
In setting up the problem, it is assumed that a beam of impurity atoms
(Phosphorus, Gallium, Indium and Arsenic) of flux density ( )120 secm −−J falls
normally on the front surface of the material target (Silicon) along the x-axis
direction where it is partly reflected and partly driven-in into the target slab. This part
is of amount ( ),secm 120
−−jA where, ( )( )TRA −= 1 stands for an equivalent
absorption coefficient that depends in general on the energy (E) of the impinging
atom. It may depend also on the absolute temperature T of the front surface, ( )TR is
the reflectivity of the front surface. It is suggested also that the impurity atoms are
supplied from an undiminished infinite source for the entire duration of diffusion
process [13]. This driven-in part will be redistributed within the target wafer.
The concentration function ( )txC , of such dopant extends till a penetration
depth ( )tδ after which this function vanishes, i.e., the diffusion process stops at this
limit.
The mathematical formulation of such a problem is given as follows:
The parabolic diffusion equation can be written in the form
( )
( ) ,0,0,,, 2 >≤≤∇=
∂
∂tdxtxJD
t
txJx (1)
where
( )secm2D is the diffusion coefficient.
Equation (1) is subjected to the following conditions
(i) At ( ) ,00,,0 == xJt
EL-ADAWI, ABDEL-GHANY and SHALABY
4
(ii) At ( ) ( ),1,0,0 0 RJtJx −==
(iii) At ( ) ,0,, →δδ= tJx and
(iv) ( ) .0, →δ tC
( )120 secm −−J is defined as the number of the incident atoms per unit area per
second, ( )( )3m, −txC is the concentration of the doped atoms at the boundary x at
time t measured from the beginning of the exposure time.
Equation (1) can be rewritten in the form
( )( )
.,1
,2
t
txJ
DtxJx ∂
∂=∇ (2)
Taking Laplace transform of equation (2) with respect to the time variable t gives
( ) ( ) ( )[ ],0,,1
,2
2
xJsxJsD
sxJdx
d−= (3)
where ( )sxJ , is the Laplace transform of the function ( )., txJ
The initial condition (i) makes it possible to write equation (3) in the form
( ) ( ).,,2
2
sxJD
ssxJ
dx
d= (4)
The solution is written in the form
( ) .,x
D
sx
D
s
BeAesxJ−
+= (5)
The Laplace transform of the boundary condition (ii) for =0J constant gives
( )( )( )
.1
,0 0
s
TRJsJ
−= (6)
Substituting (6) into (5) at ,0=x one gets
( )( )
.10
s
TRJBA
−=+ (7)
Similarly, the Laplace transform of the boundary condition (iii) gives
( ) .0, =δ sJ (8)
DIFFUSION IN SEMICONDUCTORS BY USING LAPLACE’S …
5
Substituting equation (8) into equation (5) at ,δ=x one gets
.0=+δ−δ
D
s
D
s
BeAe (9)
Solving simultaneously, equations (7) and (9), one gets
( )( ) δ−
δ
−−= D
s
e
D
ss
TRJA
sinh2
10 (10)
and
( )( )
.
sinh2
10δ+
δ
−= D
s
e
D
ss
TRJB (11)
Substituting equations (10) and (11) into equation (5), one gets the solution in the
form
( )( )( ) ( ) ( )
,
sinh2
1, 0
−
δ
−=
−δ−−δ xD
sx
D
s
ee
D
ss
TRJsxJ (12)
( )( )( )
( ) .sinh
sinh
1, 0
−δ
δ
−= x
D
s
D
ss
TRJsxJ (13)
( )txJ , is obtained by using the inverse Laplace transform of equation (13) (standard
tables [14, 15]), in the form
( )txJ ,
( )( )( ) ( )
.sin12
1
2
1
0
δ
−δπ
−
π+
δ
−δ−=
δ
π−∞
=∑ xn
en
xTRJ
tn
D
n
n
(14)
3. Determination of the Concentration Function ( ) 3m−tx,C
Considering Fick’s first law
EL-ADAWI, ABDEL-GHANY and SHALABY
6
( ) ( ).,, txCDtxJ ∇−= (15)
Thus
( )
( )
( )
( ) .,1
,
0
,
,0∫∫ −=
xtxc
tc
dxtxJD
txdC (16)
From equations (14) and (16) one gets
( ) ( )( )( )
D
TRJtCtxC
−−=
1,0, 0
( ) ( ).coscos
21
21
22
22
π−
δ
−δπ
π
δ−+
δ−× ∑
∞
=
δ
π−
nxn
en
xx
n
tn
Dn
(17)
Moreover, let us consider the balance equation for the particle flux density written in
the form
( )( ) ( ) .,1
00
0 ∫∫δ
=− dxtxCdtTRJ
t
(18)
Substituting equation (17) into equation (18), gives the expression for ( )tC ,0 in the
form
( ) ( )( )( )( )
.2
3
11
1,0
2
122
0
00
π
δ−
δ−
+−δ
=
δ
π−∞
=∑∫
tn
D
n
t
enD
TRJdtTRJtC (19)
The concentration function ( )txC , can be obtained by substituting (19) into (17) in
the form
( ) ( )( )( )( )
( ) ( ).
cos21
231
11
, 2
122
2
0
00
δ
−δπ
π
δ−
−
δ+−
δ
−+−
δ=
δ
π−∞
=∑
∫xn
en
xx
D
TRJdtTRJtxC
tn
D
n
n
t
(20)
DIFFUSION IN SEMICONDUCTORS BY USING LAPLACE’S …
7
4. Determination of the Diffusion Penetration Depth ( )mδ
Using the boundary condition (iv) at δ=x in equation (20) one gets
( )( )( )( ) ( )
.21
6
11
10
1220
00
2
π
δ−−
δ
−−
+−δ
= ∑∫∞
=
δ
π−
n
tn
Dnt
enD
TRJdtTRJ (21)
Considering the order of the terms in equation (21), one can neglect the term
( )∑∞
=
δ
π−
π
δ−
122
2
21
n
tn
Dn
en
with respect to .6
δ Thus one finally gets the diffusion
penetration depth function as
.6Dt≅δ (22)
5. Computations
Computations are carried out according to values in Table 1, considering
constant incident particle flux ,secm105 12150
−−×=J ( )kTEDD vExp0=
12 secm − and ( )TR for silicon )T51012.3322.0 −×+= [16] in the range
( ).K1685-K300
Table 1. The values of vED ,0 for the diffused elements of Phosphorus, Gallium,
Indium and Arsenic diffused into Silicon target material [17]
Parameters
Element
sec/m, 20D eV,vE
Arsenic 3.2 E-5 3.56
Phosphorus 3.0 E-4 3.68
Indium 16 E-4 3.9
Gallium 3.6 E-4 3.51
It is suggested that the relative size of the dopant atom relative to the host
medium atoms plays also an important role in the diffusion process. Table 2 gives the
size of the impurity Gallium and Indium atoms relative to Silicon [18].
EL-ADAWI, ABDEL-GHANY and SHALABY
8
Table 2. Size of some impurity atoms relative to Silicon [18]
Element Size relative to Silicon
Gallium 1.07
Indium 1.22
The electron configuration of neutral atoms in their ground states is related also to
the size of the diffused atoms.
Table 3. The outer configuration for some neutral atoms in their ground states
Element Electron configuration
14Si 22 p3s3
15P 32 p3s3
31Ga 12 p4s4
33As 3p4s4 2
49In 12 p5s5
The following special cases are considered:
(i) sec1=t
The concentration function ( )txC , against the depth x is computed for the
diffusion of Indium, Phosphorous, into Silicon (Figure 1, for 1-E313.3=R for
,K300=T and Figure 3, for 1-E5.3=R and ).K900=T The same dependence
for the diffusion of Arsenic and Gallium, into Silicon (Figure 2, for 1-E313.3=R
and ,K300=T Figure 4, for 1-E5.3=R and ).K900=T
(ii) At ( ) m10δ=x
The concentration function ( )txC , against the diffusion time t is computed for
the diffusion of Indium, Phosphorous, into Silicon (Figure 5, for ,1-E313.3=R
K300=T and Figure 7, for ).K900,1-E5.3 == TR The same dependence for
the diffusion of Arsenic and Gallium, into Silicon (Figure 6, for 1-E313.3=R and
DIFFUSION IN SEMICONDUCTORS BY USING LAPLACE’S …
9
,K300=T Figure 8, for 1-E5.3=R and ).K900=T
6. Discussions
1. The obtained expression for the concentration function ( ) 3m, −txC (equation
(20)) reveals that it depends principally on the flux density .sm, 120
−−J The
dependence is linear for constant flux density.
2. The same function depends fundamentally on the diffusion coefficient
.sm, 12 −D
3. The diffusion penetration depth depends basically on the diffusion coefficient
12sm, −D which in turn depends on the absolute temperature T. So the penetration
depth becomes greater and the concentration becomes smaller at K900=T than
that at K.300=T Such depth depends also on the size of the dopant atom relative
to the host atom, this is evident in Figures 1, 2, 3 and 4.
4. Dopants of higher relative sizes are more accumulated than dopants of smaller
relative sizes at a certain layer and at a certain diffusion time (Figures 5, 6, 7 and 8).
7. Conclusions
1. The concentration of the Indium and Arsenic dopants into Silicon at a certain
layer with time is greater than that for Phosphorous and Gallium dopants into Silicon
for the same operating conditions.
2. The penetration depth of the Indium and Arsenic are less than that for
Phosphorous and Gallium into Silicon for the same operating conditions.
References
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London, 1976.
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John-Wiley & Sons, 2004.
[4] V. Hung, P. Hong and B. Khue, Boron and Phosphorus diffusion in Silicon;
EL-ADAWI, ABDEL-GHANY and SHALABY
10
interstitial, vacancy, and combination mechanisms, Prac. Nalt. Conf. Theor. Phys. 35
(2010), 73.
[5] Q. Zhao, D. C. Zhang and J. F. Lin, Surface materials draperies condensation mode by
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Arsenic mediated inter diffusion across Germanium/Silicon interfaces, Electrochemical
and Solid State Letters 5(2) (2002), G5-G7.
[14] M. R. Speigel, Laplace Transforms, Schaum’s Outline Series, McGraw-Hill Book
Company, New York, USA, 1965.
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London, 1966.
[16] A. Battacharyya and B. G. Streetman, Dynamics of pulsed CO laser annealing of
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[17] R. B. Fair, Concentration profiles of diffuse dopants in Silicon, Impurity Dopant
Processes in Silicon, F. Y. Y. Yang, ed., North Holland, 1981.
[18] Sorab K. Ghandhi, The Theory and Practice of Micro Electronics, John Wiley, 1968.
DIFFUSION IN SEMICONDUCTORS BY USING LAPLACE’S …
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Figure 1. Shows the concentration of Indium and Phosphorus in Silicon as a function
of the depth in the Silicon target. At 1-E313.3sec,1 == Rt and K.300=T
Figure 2. Shows the concentration of Indium and Phosphorus in Silicon as a function
of the depth in the Silicon target. At 1-E3501.3sec,1 == Rt and K.900=T
EL-ADAWI, ABDEL-GHANY and SHALABY
12
Figure 3. Shows the concentration of Arsenic and Gallium in Silicon as a function of
the depth in the Silicon target. At 1-E313.3sec,1 == Rt and K.300=T
Figure 4. Shows the concentration of Arsenic and Gallium in Silicon as a function of
the depth in the Silicon target. At 1-E3501.3sec,1 == Rt and K.900=T
DIFFUSION IN SEMICONDUCTORS BY USING LAPLACE’S …
13
Figure 5. Shows the time dependence of the concentration of Indium and Phosphorus
in Silicon target. At 1-E313.3m,10 =δ= Rx and K.300=T
Figure 6. Shows the time dependence of the concentration of Indium and Phosphorus
in Silicon target. At 1-E3501.3m,10 =δ= Rx and K.900=T
EL-ADAWI, ABDEL-GHANY and SHALABY
14
Figure 7. Shows the time dependence of the concentration of Arsenic and Gallium in
Silicon target. At 1-E313.3m,10 =δ= Rx and K.300=T
Figure 8. Shows the time dependence of the concentration of Arsenic and Gallium in
Silicon. 1-E3501.3m,10 =δ= Rx and K.900=T
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